(CNN) -- Airlines canceled more than four thousand flights Tuesday as a massive snowstorm of historic proportions coated the nation's heartland with a thick blanket of snow.

"It's going to take days to clear out this storm," said CNN meteorologist Jacqui Jeras. "This is a monster."

With more than 30 states under winter storm warnings or blizzard warnings, air traffic has come to a halt in some places. Dallas and Houston in Texas and Chicago, Illinois, are feeling the most impact, said FAA spokeswoman Tammy Jones.

"Travel will likely be difficult to impossible in many areas," the National Weather Service said.

Up to two feet of snow is expected to fall in the Chicago area, where some weary travelers were already frustrated about not being able to continue their journey.

"I just want to get on a plane today... and get out of here. I don't want to be stuck here," said Kristine Whitcher, a passenger from Canada who was traveling through O'Hare International.

She's not likely to get her wish very soon.

More than 1,300 flights were canceled at the airport on Tuesday. Few flights, if any, are expected to operate out of O'Hare on Wednesday, said Karen Pride, a spokeswoman for the Chicago Department of Aviation. United Airlines, Continental Airlines and American Airlines are suspending operations at the airport after 6 p.m. ET on Tuesday.

At Midway, airlines have canceled flight operations for the day. The carriers expect to begin operating again on Wednesday afternoon, Pride said.

Both Chicago airports are open and ready to implement a passenger assistance program if the storm leaves any travelers stranded. Hundreds of cots will be available and some concessions will remain open to accommodate them, Pride said.

It was also slow going at other airports.

Dallas-Fort Worth International reopened late Tuesday morning after shutting down for several hours because of ice, spokesman David Magana said. Only one of seven runways is open and officials are expecting hundreds of flight cancellations.

"The biggest problem we have here at DFW are the winds," said American Airlines spokesman Ed Martelle. "We can't put our guys up there in the in de-icing buckets with 45 mph winds."

American Airlines canceled about 1,900 flights for Tuesday, 800 of those at Dallas-Forth Worth alone. The carrier is also scrubbing more than 1,000 flights for Wednesday.

AirTran passengers traveling through two dozen cities in the region -- including Chicago, St. Louis and Boston -- can change their reservation without penalty.

Delta Air Lines canceled 625 flights for Tuesday.

Ahead of the storm, Delta issued a winter weather advisory for the northern United States. Passengers traveling to airports in 20 states through Friday can reschedule their itineraries without a fee or get a refund if their flights are canceled or significantly delayed.

Continental Airlines, which canceled 650 flights, will allow changes for travel to, through and from nearly three dozen airports, including the airline's hubs in Newark, New Jersey, and Cleveland, Ohio, for travelers scheduled to fly through Thursday.

US Airways has also relaxed its change-fee policies for passengers scheduled to fly through Thursday to or from 27 cities from Portland, Maine, to Akron, Ohio.

United Airlines, which canceled 800 flights, has issued travel waivers for passengers traveling to, from or through 19 states in the Midwest and Northeast as well as five Canadian cities. Travelers will be able to alter their plans without fees or get a full refund if their flight is canceled.

American Airlines is allowing travelers who are scheduled to fly on Monday through Thursday from more than 30 airports in the Midwest -- including those in Chicago, Detroit and Milwaukee, Wisconsin -- to change their plans without penalty.

Despite all of the flight cancellations, many people are still on the move.

Train travel remains largely unaffected by the storm -- at least for now. Save for some "minor service adjustments" north of New York City and between New York and Boston, Massachusetts, Amtrak trains are running on schedule.